Due to the unprecedented brilliance of the beams, the performance of the Compact Linear Collider (CLIC) damping rings (DR) is affected by collective effects. Single bunch instability thresholds based on a broad-band resonator model and the associated coherent tune shifts have been evaluated with the HEADTAIL code. Simulations performed for positive and negative values of chromaticity showed that higher order bunch modes can be potentially dangerous for the beam stability. This study also includes the effects of high frequency resistive wall impedance due to different coatings applied on the chambers of the wigglers for e-cloud mitigation and/or ultra-low vacuum pressure. The impact of the resistive wall wake fields on the transverse impedance budget is finally discussed.